Q.Molecules in air in the atmosphere are attracted by gravitational force of the earth. Explain why all of them do not fall into the earth just like an apple falling from a tree.

The key idea is the kinetic theory of gases: air molecules are in constant, random thermal motion with speeds far exceeding their terminal fall speed due to gravity.

1. Each molecule has a root-mean-square speed given by $v_{\text{rms}} = \sqrt{\frac{3kT}{m}}$. At typical atmospheric temperatures, this speed is hundreds of metres per second.
2. Gravity does pull molecules downward, but the thermal kinetic energy ($\frac{3}{2}kT$) is much larger than the gravitational potential energy change over a molecular mean free path. A molecule's downward drift from gravity is negligible compared to its random, high-speed motion. …

The key idea here is a battle between two tendencies: gravity wants to pull every molecule down to the surface, while the random thermal motion of the molecules (their kinetic energy) wants to spread them out evenly in all directions. An apple falls because its thermal energy is tiny compared to its gravitational potential energy — but for a single air molecule, the situation is very different.

Let’s walk through the physics step by step.

1. Gravity acts on every molecule, just as it does on an apple.

   Each air molecule (mostly nitrogen and oxygen) has mass, so Earth’s gravity exerts a downward force $mg$ on it. If molecules were stationary, they would indeed fall straight down, just like a dropped apple. But they are not stationary.

2. Molecules are in constant random motion due to temperature.

   Air at room temperature has an average molecular speed of about $500\ \text{m/s}$. This is thermal motion — the molecules are zipping around in all directions, colliding with each other and with surfaces. The average kinetic energy per molecule is $\frac{3}{2} k_B T$, where $k_B$ is Boltzmann’s constant and $T$ is the absolute temperature.

3. Compare the thermal energy to the gravitational potential energy.

   Consider a molecule at a height $h$ above the ground. Its gravitational potential energy is $mgh$. For the molecule to be “trapped” near the ground, $mgh$ must be much larger than its thermal energy $k_B T$. Let’s check typical numbers:

   • Mass of an $\text{N}_2$ molecule: $m \approx 4.65 \times 10^{-26}\ \text{kg}$
   • $g = 9.8\ \text{m/s}^2$
   • $k_B = 1.38 \times 10^{-23}\ \text{J/K}$
   • $T \approx 300\ \text{K}$ The thermal energy is $k_B T \approx 4.14 \times 10^{-21}\ \text{J}$. The gravitational potential energy at, say, $h = 1\ \text{m}$ is $mgh \approx 4.56 \times 10^{-25}\ \text{J}$. That’s ten thousand times smaller than the thermal energy! So a molecule at 1 m height has far more kinetic energy than the gravitational pull can overcome — it easily flies upward again.

4. The atmosphere is in a dynamic equilibrium, not a static pile. The molecules are constantly moving up and down. Gravity does pull them down, but thermal motion kicks them back up. The result is that the density of air decreases exponentially with height — this is the barometric formula: …

Step 1: gravity pulls molecules down; thermal motion (avg KE ~(3/2)kT) spreads them out. Step 2: for a N2 molecule at h=1m, mgh 4.6e-25 J vs kT4.1e-21 J at 300K -- thermal energy dominates by ~4 orders of magnitude. Step 3: molecules constantly rebound via collisions, settling into the barometric distribution n(h)=n0*exp(-mgh/kT), decaying over kilometres. Step 4: contrast a macroscopic apple ( …